Onsager reciprocity theorem

In this framework, the intensity of an entropy source is represented by a quadratic form of thermodynamic forces. The corresponding phenomenological coefficients form a matrix with remarkable properties. These properties, formulated as the Onsager reciprocity theorem, allow to reduce the number of independent quantities and to find relations between various physical effects. 

Hurley, J. Garrod, C. (1982). Generalization of Onsager reciprocity theorem. Physical Review Letters, 48 (23), pp. 1575. 

Ozer, M., Provaznik, I., 2005. A comparative tool for the validity of rate kinetics in ion channels by Onsager reciprocity theorem. J. Theor. Biol. 233, 237-243. 

Unlike other branches of physics, thermodynamics in its standard postulation approach [272] does not provide direct numerical predictions. For example, it does not evaluate the specific heat or compressibility of a system, instead, it predicts that apparently unrelated quantities are equal, such as (1 A"XdQ/dP)T = - (dV/dT)P or that two coupled irreversible processes satisfy the Onsager reciprocity theorem (L 2 L2O under a linear optimization [153]. Recent development in both the many-body and field theories towards the interpretation of phase transitions and the general theory of symmetry can provide another plausible attitude applicable to a new conceptual basis of thermodynamics, in the middle of Seventies Cullen suggested that thermodynamics is the study of those properties of macroscopic matter that follows from the symmetry properties of physical laws, mediated through the statistics of large systems [273], It is an expedient happenstance that a conventional simple systems , often exemplified in elementary thermodynamics, have one prototype of each of the three characteristic classes of thermodynamic coordinates, i.e., (i) coordinates conserved by the continuous space-time symmetries (internal energy, U), (ii) coordinates conserved by other symmetry principles (mole number, N) and (iii) non-conserved (so called broken ) symmetry coordinates (volume, V). 

The usual emphasis on equilibrium thermodynamics is somewhat inappropriate in view of the fact that all chemical and biological processes are rate-dependent and far from equilibrium. The theory of non-equilibrium or irreversible processes is based on Onsager s reciprocity theorem. Formulation of the theory requires the introduction of concepts and parameters related to dynamically variable systems. In particular, parameters that describe a mechanism that drives the process and another parameter that follows the response of the systems. The driving parameter will be referred to as an affinity and the response as a flux. Such quantities may be defined on the premise that all action ceases once equilibrium is established. 

Onsager s theorem deals with reciprocal relations in irreversible resistive processes, in the absence of magnetic fields [114], The resistive qualifier signifies that the fluxes at a given instant depend only on the instantaneous values of the affinities and local intensive parameters at that instant. For systems of this kind two independent transport processes may be described in terms of the relations... 

Onsager s theorem consists of proving that a reciprocal relationship of the type Lap = Lpa between the affinities and fluxes of coupled irreversible processes is universally valid in the absence of magnetic fields. 

G. Gallavotti, Chaotic hypothesis Onsager reciprocity and fluctuation dissipation theorem. J. Stat. Phys. 84, 899 (1996). 

The kinetic constants of the system enter into the phenomenological L-coefficients, which are parameters of state. According to the reciprocity theorem of Onsager, the cross-coefficients L+r and Lr+ are identical. Now the definition of the efficiency 17 emerges directly from the dissipation function... 

From a satisfactory, to a certain extent, explanation based on the second law of the Prigogine theorem we can pass to an absolutely macroscopic explanation of the Onsager reciprocal relations by changing the order of proofs accepted in the nonequilibrium thermodynamics (in the nonequilibrium thermodynamics the Prigogine theorem is derived from the Onsager relations). 

Fortunately, several simplifications can be made (Nye, 1957). Transport phenomena, for example, are processes whereby systems transition from a state of nonequilibrium to a state of equilibrium. Thus, they fall within the realm of irreversible or nonequilibrium thermodynamics. Onsager s theorem, which is central to nonequilibrium thermodynamics, dictates that as a consequence of time-reversible symmetry, the off-diagonal elements of a transport property tensor are symmetrical (i.e., xy = X/,-). This is known as a reciprocal relation. The Norwegian physical chemist Lars Onsager (1903-1976) was awarded the 1968 Nobel Prize in Chemistry for reciprocal relations. Thus, the tensor above can be rewritten as... 

According to Onsager s reciprocity theorem Cl], the matrix of the phenomenological equations (7) is symmetric and we have... 

We should also mention that the normal solution of the Boltzmann equation discussed here, together with the //-theorem discussed in the previous section, can be used to provide a derivation of the principles of nonequilibrium thermodynamics. For mixtures, one can show that the various diffusion coefficients that occur in the Navier-Stokes equations can be expressed in a form where Onsager reciprocal relations are satisfied. However, both for mixtures and for pure gases the relation between the normal solution and irreversible thermodynamics only holds if one does not go beyond in the -expansion of the distribution function. ... 

The minimum entropy production theorem dictates that, for a system near equilibrium to achieve a steady state, the entropy production must attain the least possible value compatible with the boundary conditions. Near equilibrium, if the steady state is perturbed by a small fluctuation (8), the stability of the steady state is assured if the time derivative of entropy production (P) is less than or equal to zero. This may be expressed mathematically as dPIdt 0. When this condition pertains, the system will develop a mechanism to damp the fluctuation and return to the initial state. The minimum entropy production theorem, however, may be viewed as providing an evolution criterion since it implies that a physical system open to fluxes will evolve until it reaches a steady state which is characterized by a minimal rate of dissipation of energy. Because a system on the thermodynamic branch is governed by the Onsager reciprocity relations and the theorem of minimum entropy production, it cannot evolve. Yet as a system is driven further away from equilibrium, an instability of the thermodynamic branch can occur and new structures can arise through the formation of dissipative structures which requires the constant dissipation of energy. 

Perhaps the best starting point in a review of the nonequilibrium field, and certainly the work that most directly influenced the present theory, is Onsager s celebrated 1931 paper on the reciprocal relations [10]. This showed that the symmetry of the linear hydrodynamic transport matrix was a consequence of the time reversibility of Hamilton s equations of motion. This is an early example of the overlap between macroscopic thermodynamics and microscopic statistical mechanics. The consequences of time reversibility play an essential role in the present nonequilibrium theory, and in various fluctuation and work theorems to be discussed shortly. 

We can describe irreversibility by using the kinetic theory relationships in maximum entropy formalism, and obtain kinetic equations for both dilute and dense fluids. A derivation of the second law, which states that the entropy production must be positive in any irreversible process, appears within the framework of the kinetic theory. This is known as Boltzmann s H-theorem. Both conservation laws and transport coefficient expressions can be obtained via the generalized maximum entropy approach. Thermodynamic and kinetic approaches can be used to determine the values of transport coefficients in mixtures and in the experimental validation of Onsager s reciprocal relations. 

Onsager (1931) in his celebrated theorem on the reciprocal relations, was able to show that, as long as the forces and flows appearing in Eq. (13.4.1) are obtained in such a way that Eq. (13.3.11) is valid, and the forces are linearly independent, the phenomenological coefficients L t satisfy the relation... 

In the following we are going to introduce the generalized Onsager constitutive theory for a non-linear system of constitutive (9) satisfying the (18) reciprocity relations, the (13) equilibrium conditions and the (10) second law of thermodynamics. In the following we shall present that the Edelen s decomposition theorem [4] is valid in every class of the thermodynamic forces, which are two times continuously differentiable with respect to fluxes. 

Some of the variational principles which are to be described in the present article, are very closely connected with Onsager s reciprocity relationsAlthough there have been various methods of derivation of this theorem, we shall follow the traditional method of derivation by Onsager and Casimir. This is based on the consideration of fluctuations in an aged system, and this method is also connected with the derivation of Onsager s principle of least dissipation of energy. ... 

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